Let $h_n$ denote the class number of $n$-th layer of the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}$. Weber proved that $h_n\;\;(n\ge 1)$ is odd and Horie proved that $h_n\;\;(n\ge 1)$ is not divisible by a prime number $\ell $ satisfying $\ell \equiv 3,\,5\hspace{4.44443pt}(\@mod \; 8)$. In a previous paper, the authors showed that $h_n\;\;(n\ge 1)$ is not divisible by a prime number $\ell $ less than $10^7$. In this paper, by investigating properties of a special unit more precisely, we show that $h_n\;\;(n\ge 1)$ is not divisible by a prime number $\ell $ less than $1.2\cdot 10^8$. Our argument also leads to the conclusion that $h_n\;\;(n\ge 1)$ is not divisible by a prime number $\ell $ satisfying $\ell \lnot \equiv \pm \;1\hspace{4.44443pt}(\@mod \; 16)$.